1. Chapter 10: Algorithm Design Techniques
Mark Allen Weiss: Data Structures and Algorithm Analysis in Java
Chapter 10: Algorithm Design Techniques
Lydia Sinapova, Simpson College

2. Algorithm Design Techniques
Brute Force
Greedy Algorithms
Divide and Conquer
Dynamic Programming
Transform and Conquer
Backtracking
Genetic Algorithms

3. Brute Force
Based on the problem’s statement and definitions of the concepts involved.
Examples:
Sequential search
Exhaustive search: TSP, knapsack
Simple sorts: selection sort, bubble sort
Computing n!

4. Greedy Algorithms "take what you can get now" strategy Work in phases.
In each phase the currently best decision is made

5. Greedy Algorithms - Examples
Dijkstra's algorithm
(shortest path is weighted graphs)
Prim's algorithm, Kruskal's algorithm
(minimal spanning tree in weighted graphs)
Coin exchange problem
Huffman Trees

6. Divide and Conquer
Reduce the problem to smaller problems (by a factor of at least 2) solved recursively and then combine the solutions
Examples: Binary Search
Mergesort
Quick sort
Tree traversal
In general, problems that can be defined recursively

7. Decrease and Conquer
Reduce the problem to smaller problems solved recursively and then combine the solutions
Examples of decrease-and-conquer algorithms:
Insertion sort Topological sorting Binary Tree traversals: inorder, preorder and postorder (recursion) Computing the length of the longest path in a binary tree (recursion) Computing Fibonacci numbers (recursion) Reversing a queue (recursion)

8. Dynamic Programming
Bottom-Up Technique in which the smallest sub-instances are explicitly solved first and the results of these used to construct solutions to progressively larger sub-instances.
Example:
Fibonacci numbers computed by iteration.

9. Transform and Conquer
The problem is modified to be more amenable to solution. In the second stage the problem is solved.
Problem simplification e.g. presorting Example: finding the two closest numbers in an array of numbers. Brute force solution: O(n2) Transform and conquer with presorting: O(nlogn)
Change in the representation Example: AVL trees guarantee O(nlogn) search time
Problem reduction Example: least common multiple: lcm(m,n) = (m*n)/ gcd(m,n)

10. Backtracking Based on exhaustive search in multiple choice problems
Generate-and-Test methods
Based on exhaustive search in multiple choice problems
Typically used with depth-first state space search problems.
Example: Puzzles

11. Backtracking – State Space Search
initial state
goal state(s)
a set of intermediate states
a set of operators that transform one state into another. Each operator has preconditions and postconditions.
a cost function – evaluates the cost of the operations (optional)
a utility function – evaluates how close is a given state to the goal state (optional)

12. Genetic Algorithms Search for good solutions among possible solutions
The best possible solution may be missed
A solution is coded by a string , also called chromosome. The words string and chromosome are used interchangeably
A strings fitness is a measure of how good a solution it codes. Fitness is calculated by a fitness function
Selection: The procedure to choose parents
Crossover is the procedure by which two chromosomes mate to create a new offspring chromosome
Mutation : with a certain probability flip a bit in the offspring

13. Basic Genetic Algorithm
Start: Generate random population of n chromosomes (suitable solutions for the problem)
Fitness: Evaluate the fitness f(x) of each chromosome x in the population
New population: Create a new population by repeating following steps until the new population is complete
Test: If the end condition is satisfied, stop, and return the best solution in current population

14. New Population
Selection: Select two parent chromosomes from a population according to their fitness
Crossover: With a crossover probability cross over the parents to form a new offspring (children). If no crossover was performed, offspring is an exact copy of parents.
Mutation: With a mutation probability mutate new offspring at each locus (position in chromosome).

15. Conclusion How to choose the approach?
First, by understanding the problem, and second, by knowing various problems and how they are solved using different approaches.
Strongly recommended reading to learn more about how to design algorithms:
The Algorithm Design Manual, Steven S. Skiena Department of Computer Science, State University of New York
Algorithm Design Paradigms, Paul E. Dunne
University of Liverpool, Department of Computer Science